Rank one perturbations and Anderson-type Hamiltonians

Abstract

Motivated by applications of the discrete random Schrödinger operator, mathematical physicists and analysts, began studying more general Anderson-type Hamiltonians; that is, the family of self-adjoint operators $$H_{\omega }=H+V_{\omega }$$ on a separable Hilbert space $\mathcal{H}$, where the perturbation is given by $$V_{\omega }=\sum _n{\omega }_n(\cdot ,{\phi }_n){\phi }_n$$ with a sequence $\{{\phi }_n\}\subset \mathcal{H}$ and independent identically distributed random variables ${\omega }_n$. We show that the the essential parts of Hamiltonians associated to any two realizations of the random variable are (almost surely) related by a rank one perturbation. This result connects one of the least trackable perturbation problem (with almost surely non-compact perturbations) with one where the perturbation is `only' of rank one perturbations. The latter presents a basic application of model theory. We also show that the intersection of the essential spectrum with open sets is almost surely either the empty set, or it has non-zero Lebesgue measure.

14 pages

References in corpus (3)

• Rank one perturbations and singular integral operators
• A new numerical approach to Anderson (de)localization
• Reverse Carleson Embeddings for Model Spaces

Cited by in corpus (3)

• Delocalization for the 3-D discrete random Schroedinger operator at weak disorder
• Rank one and finite rank perturbations - survey and open problems
• Spectral Analysis of Iterated Rank-One Perturbations